Ex 5.1, 10

Transcript

Ex 5.1, 10 Find all points of discontinuity of f, where f is defined by 𝑓 𝑥﷯= 𝑥+1, 𝑖𝑓 𝑥≥1﷮&𝑥2+1 , 𝑖𝑓 𝑥<1﷯﷯ We have 𝑓 𝑥﷯= 𝑥+1, 𝑖𝑓 𝑥≥1﷮&𝑥2+1 , 𝑖𝑓 𝑥<1﷯﷯ Case 1: At x = 1 f is continuous at x = 1 y L.H.L = R.H.L = 𝑓 1﷯ i.e. lim﷮x→ 1﷮−﷯﷯ 𝑓 𝑥﷯ = lim﷮x→ 1﷮+﷯﷯ 𝑓 𝑥﷯ = 𝑓 1﷯ 𝑓 𝑥﷯=𝑥+1 𝑓 1﷯=1+1 = 2 Thus, L.H.L = R.H.L = 𝑓 1﷯ f is continuous at x = 1 Case 2 Let x = c (where c < 1) ∴ 𝑓 𝑥﷯= 𝑥﷮2﷯+1 f is continuous at x = c if if lim﷮x→𝑐﷯ 𝑓 𝑥﷯=𝑓(𝑐) Thus , lim﷮x→𝑐﷯ 𝑓 𝑥﷯=𝑓(𝑐) ⇒ f is continuous for 𝑥 =𝑐 ( c is less than 1 ) ⇒ f is continuous for all real numbers less than 1. Case 3 Let x = c (where c > 1) 𝑓 𝑥﷯=𝑥+1 f is continuous at x = c if lim﷮x→𝑐﷯ 𝑓 𝑥﷯=𝑓(𝑐) Thus lim﷮x→𝑐﷯ 𝑓 𝑥﷯=𝑓(𝑐) ⇒ f is continuous at 𝑥 =𝑐 ( c is less than 1) ⇒ f is continuous at all real numbers less than 1 Hence, there is no point of discontinuity ⇒ f is continuous at all real point. Thus, f is continuous for all 𝒙∈𝐑.